1. What are visual explanations?

Perhaps the most famous and certainly one of the oldest visual
explanations in mathematics is this visual proof of the Pythagorean
theorem.

The theorem states that the yellow square (``the square on the
hypothenuse'') is equal to the green
square plus the blue square; this should become
clear upon contemplation of the two compound squares on the right.

This proof is unusual in its brevity and its complete appropriateness
to the problem. Pictures and diagrams are also used in non-geometrical
parts of mathematics, mostly for psychological reasons: harnessing our
ability to reason ``visually'' with the elements of a diagram in
order to assist our more purely logical or analytical thought processes.

Some diagrams are more useful than others. Edward R. Tufte has
written three books analyzing the effectiveness of figures and
diagrams in communicating ideas. His latest,Visual Explanations -Images
and Quantities, Evidence and Narrative (Graphics Press, Cheshire CT 1997)
was reviewed in the
January 1999
AMS Notices by
Bill Casselman of the University of British Columbia. Casselman focusses on
the applications of Tufte's ideas to explanations in mathematics, and
distills from the book a set of rules designed to make graphics
contribute most effectively to the communication of mathematics.

Tufte's Rules, after Casselman, abridged:

Tone down secondary elements of a picture: layer
the figure to produce a visual hierarchy.

Replace coded labels in the figure by direct ones.

Produce emphasis by using the smallest possible effective
distinctions.

Eliminate all unnecessary parts of a figure.

Use small multiples: numerous repetitions of a single
figure with slight variations.

Make the graphics carry a story.

Casselman gives a sample application of these rules to a visual
proof of the incommensurability of side and diagonal in a
regular pentagon. This proposition is of additional interest
because the ratio of these two lengths is exactly the Golden Mean.
In this column we
will follow an adaptation of Casselman's argument to the web, using
Casselman's own figures (thanks, Bill!)